ICF13B

displacement and magnetic induction intensity factors denoted respectively 1 2 , , D k k k and Bk . To the best of the authors’ knowledge, this problem was not considered in the open literature to-date. Problem description and formulation As shown in Figure 1, the problem under consideration consists of a functionally graded magneto electro elastic layer containing an embedded crack of length 2a along the x-axis. The crack surfaces are assumed to be partial magneto electrically permeable using the magnetic and electric permeability parameters km and ke varying in between 0 and 1 representing the cases of completely impermeable and completely permeable crack surfaces, respectively. Consequently, crack faces are subjected to mechanical tangential and normal tractions 1(x) and 2(x), electric displacement (1-ke)E(x), and magnetic induction (1-km)B(x). The graded layer is modeled as a nonhomogeneous elastic medium with magneto electromechanical properties varying in the depth direction (y-coordinate) as follows:     11 13 33 44 110 130 330 440 , , , , , , e , y c c c c c c c c       15 31 33 150 310 330 , , , , e , y e e e e e e   ,y (1a,b)     15 31 33 150 310 330 , , , , e , y f f f f f f       11 33 110 330 , , e , y      ,y (1c,d)     11 33 110 330 , , e , y g g g g       11 33 110 330 , , e , y      .y (1e,f) where 0 0 0 0 0 0 , , , , , ij ij ij ij ij ij c e f g   are the value of the magneto electromechanical coefficient in the FGMEEM layer along the axis 0 y  and  is the nonhomogeneity parameter controlling the variation of these coefficient in the graded layer. Figure 1. Geometry and loading of the crack problem Neglecting body forces and local electric charge, assuming small deformations and considering linear constitutive laws, the basic equations consisting of equilibrium equations and Gauss’s laws for electricity and magnetism can be combined, resulting in the following governing magneto electro elasticity equations: 2 2 2 2 2 2 2 2 11 33 12 33 21 13 21 13 2 2 0, u u u v v v c c c c e e f f x y y x y x y x x y x y x x y x y x                                                                             (2a) 2 2 2 2 2 2 2 2 33 12 33 22 13 22 13 22 2 2 2 2 2 2 0, u u u v v v c c c c e e f f x y x y x x y y x y y x y y                                                                         (2b) 2 2 2 2 2 2 2 2 13 21 13 22 11 22 11 22 2 2 2 2 2 2 0, u u u v v v e e e e g g x y x y x x y y x y y x y y                                                                           (2c)

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